Page 38 -

§5. The Group Law on Cubic Curves and Elliptic Curves 15

In all cases the first question is what can be said about the group E(k) where
E is an elliptic curve over k. The first chapter of the book is devoted to looking at
examples of groups E(k). Now we can restate Mordell’s theorem in a more natural
form.
(5.2) Theorem (Mordell 1921). Let E be a rational elliptic curve. The group of
rational points E(Q) is a finitely generated abelian group.
A rational elliptic curve is an elliptic curve defined over the rational numbers.
The proof of this theorem will be given in Chapter 6 and is one of the main results
in this book. The result was, at least implicitly, conjectured by Poincar´ e [1901] in
Sur les Properi´ etes Arithem´ etiques des Courbes Alg´ ebriques, where he defined the
rank of an elliptic curve over the rationals as the rank of the abelian group E(Q).
He studied the properties of the rank in terms of which elements are of the form 3P.
Mordell in his proof looked at the rank in terms of which elements are of the form 2P
and then substracted off from a given element R elements of the form 2P to arrive at
a finite set of generators. This is a descent procedure which goes back to Fermat.
In order to perform calculations with specific elliptic curves, it is convenient to
put the cubic equation in a standard form. In 2 we show how, by changes of variable,
3
2
2
we can eliminate three terms, y , xy ,and wx , from the ten-term general cubic
equation given at the beginning of §4 and further normalize the coefficients of x 3
2
and wy to be one. The resulting equation is called an equation in normal form (or
generalized Weierstrass equation)
2
2
2
2
3
3
wy + a 1 wxy + a 3 w y = x + a 2 wx + a 4 w x + a 6 w .
It has only one point of intersection with the line at infinity namely (0,0,1). In the x,
y-plane the equation takes the form
2
3
2
y + a 1 xy + a 3 y = x + a 2 x + a 4 x + a 6 ,
and it is this equation which is used for an elliptic curve throughout this book. If
x has degree 2 and y has degree 3 in the graded polynomial, then the equation has
weight 6 when a i has weight i. The point at infinity (0,0,1) is the zero of the group
and the lines through this zero in the x, y-plane are exactly the vertical lines. This
zero has the property that OO = O in terms of the chord-tangent composition so
that three points add to zero in the elliptic curve if and only if they lie on a line in
the plane of the cubic curve. In Chapter 1 we use this group law to calculate with an
extensive number of examples.
For an elliptic curve E over Q we can apply the structure theorem for finitely gen-
g
erated abelian groups to E(Q) to obtain a decomposition E(Q) = Z ⊕ Tors E(Q),
where g is an integer called the rank of E and Tors E(Q) is a finite abelian group
consisting of all the elements of finite order in E(Q).
In 5 we study the torsion subgroup Tors E(Q) and see that it is effectively com-
putable. From elementary consideratons related to the implicit function theorem one
can see that the group of real points E(R) is either a circle group or the circle group